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Name for "Boolean algebra (logic)"

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  • Vaughan Pratt
    What should a Wikipedia article on the syntactic (equational deduction) aspects of Boolean algebra be called? Currently there are a number of articles related
    Message 1 of 7 , Jul 10, 2007
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      What should a Wikipedia article on the syntactic (equational deduction)
      aspects of Boolean algebra be called?

      Currently there are a number of articles related in one way or another
      to Boolean algebra. It is currently proposed on the Boolean algebra
      discussion page to have two articles, one on Boolean algebras (the
      objects of the variety) which already exists under the title "Boolean
      algebra" (not as fast-paced or comprehensive as Donald Monk's very nice
      article at http://plato.stanford.edu/entries/boolalg-math/), and one on
      more syntactic aspects (relationships between operation bases,
      equational reasoning, etc.). Such a split is reasonable to permit more
      material overall without the individual articles getting too unwieldy.

      Possible names for the syntactic article include

      1. Boolean logic

      2. Boolean algebra (logic)

      3. Elementary Boolean algebra

      4. Symbolic Boolean algebra

      1 and 3 already exist as Wikipedia titles, at respectively
      http://en.wikipedia.org/wiki/Boolean_logic and
      http://en.wikipedia.org/wiki/Elementary_Boolean_algebra . No one seems
      particularly happy with 1. I wrote 3 with an eye to eventually
      substituting it for at least some of 1. The name 2 has been proposed by
      some, to go along with a renaming of "Boolean algebra" to "Boolean
      algebra (structure)" by way of disambiguation.

      What would be your preferred name for the syntactic article (not
      necessarily one of the above four)? A matching name for the algebras
      article would also be nice, but the main question for now is with the other.

      Vaughan Pratt
    • Vaughan Pratt
      I forgot to say that a common though not consistently used Wikipedia convention for disambiguating two topics naturally named the same is a parenthetical
      Message 2 of 7 , Jul 10, 2007
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        I forgot to say that a common though not consistently used Wikipedia
        convention for disambiguating two topics naturally named the same is a
        parenthetical disambiguator, as in

        Ring (mathematics)
        Ring (computer security)

        So one possible matched pair in that format could be

        Boolean algebra (elementary)
        Boolean algebra (abstract)

        Or perhaps "Boolean algebras (variety)" for the second.

        Also, while new names (if good) are helpful, what won't be helpful is to
        end up with a list of names each with exactly one expression of support.

        Vaughan
      • Brian Davey
        I vote for Boolean algebra (logic) . B ____________________________________________ Dr Brian A. Davey Reader and Associate Professor Department of
        Message 3 of 7 , Jul 10, 2007
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          I vote for "Boolean algebra (logic)".
           
          \B

          ____________________________________________

           Dr Brian A. Davey
           Reader and Associate Professor
           Department of Mathematics
           La Trobe University
           Victoria 3086
           Australia
           Phone: +61 3 9479 2599 (Office)   +61 3 9479 2600 (Sec.)
           FAX: +61 3 9479 2466
           Email: B.Davey@...
           http://www.latrobe.edu.au/www/mathstats/staff/davey/
          ____________________________________________

           


          From: univalg@yahoogroups.com [mailto:univalg@yahoogroups.com] On Behalf Of Vaughan Pratt
          Sent: Tuesday, 10 July 2007 6:42 PM
          To: univalg@yahoogroups.com
          Subject: [univalg] Name for "Boolean algebra (logic)"

          What should a Wikipedia article on the syntactic (equational deduction)
          aspects of Boolean algebra be called?

          Currently there are a number of articles related in one way or another
          to Boolean algebra. It is currently proposed on the Boolean algebra
          discussion page to have two articles, one on Boolean algebras (the
          objects of the variety) which already exists under the title "Boolean
          algebra" (not as fast-paced or comprehensive as Donald Monk's very nice
          article at http://plato. stanford. edu/entries/ boolalg-math/), and one on
          more syntactic aspects (relationships between operation bases,
          equational reasoning, etc.). Such a split is reasonable to permit more
          material overall without the individual articles getting too unwieldy.

          Possible names for the syntactic article include

          1. Boolean logic

          2. Boolean algebra (logic)

          3. Elementary Boolean algebra

          4. Symbolic Boolean algebra

          1 and 3 already exist as Wikipedia titles, at respectively
          http://en.wikipedia .org/wiki/ Boolean_logic and
          http://en.wikipedia .org/wiki/ Elementary_ Boolean_algebra . No one seems
          particularly happy with 1. I wrote 3 with an eye to eventually
          substituting it for at least some of 1. The name 2 has been proposed by
          some, to go along with a renaming of "Boolean algebra" to "Boolean
          algebra (structure)" by way of disambiguation.

          What would be your preferred name for the syntactic article (not
          necessarily one of the above four)? A matching name for the algebras
          article would also be nice, but the main question for now is with the other.

          Vaughan Pratt

        • Prohle Peter
          ... I have the following opinion based on the citations above. (A) I think, 1. Boolean logic would be one of the right titles, since it is an important
          Message 4 of 7 , Jul 11, 2007
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            > 1. Boolean logic
            > 2. Boolean algebra (logic)

            > 1 and 3 already exist as Wikipedia titles, at respectively
            > http://en.wikipedia.org/wiki/Boolean_logic and
            > http://en.wikipedia.org/wiki/Elementary_Boolean_algebra .

            > No one seems particularly happy with 1. I wrote 3 with an eye to
            > eventually substituting it for at least some of 1.

            > The name 2 has been proposed by some, to go along with a renaming of
            > "Boolean algebra" to "Boolean algebra (structure)" by way of
            > disambiguation.

            I have the following opinion based on the citations above.

            (A) I think, "1. Boolean logic" would be one of the right titles, since it
            is an important attribute of a particular logic, whether a statement can
            have one complement only or more (non-distributive logic).

            (B) Pragmatically accepting that one the right titles in question is
            already occupied, it is a bright suggestion to use

            "2. Boolean algebra (logic)" and

            "?. Boolean algebra (structure)"

            in parallel for the two subtopics in question.

            Perphaps, in the case of "2. Boolean algebra (logic)" the much more
            precise "Boolean logic" should be used as a subtitle or topic
            classification, in order to help the diverse search engines to find the
            right page.

            Peter.
          • Jorge Petrúcio Viana
            Roughly speaking... (I mean, disregarding a lot of subtle points) Bourbaky classified the mathematical structures in three categories: algebraic, order, and
            Message 5 of 7 , Jul 13, 2007
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              Roughly speaking... (I mean, disregarding a lot of subtle points)

              Bourbaky classified the mathematical structures in three categories: algebraic, order, and topological.
              Beziau introduced a fourth category: logical.

              Boolean algebras can be seen, without pain, as living inside the four universes above.
              Moreover, there are easy ways to transform a BA viewed as an algebraic structure into a BA viewed as an oder structure into a BA viewed as a topological structure into BA viewed as a logical structure, and back.
              To me, seems that algebraic, order and topological BAs are closer to each other than to logical BAs.
              So I vote with Peter:

              1. Boolean algebra (structure)

              2. Boolean algebra (logic)

              best regards,
              --
              Petrucio Viana

              Departamento de Análise
              Instituto de Matemática
              Universidade Federal Fluminense

              e-mail: petrucio@...
              web page: http://www.cos.ufrj.br/~petrucio

            • Vaughan Pratt
              ... Initially I was sceptical that these constituted distinct notions---my reaction was, aren t algebraic, order, and logical BAs all the same? (Topological
              Message 6 of 7 , Jul 15, 2007
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                > Posted by: "Jorge Petrúcio Viana" petrucio@...
                > Fri Jul 13, 2007 4:23 am (PST)
                >
                > To me, seems that algebraic, order and topological BAs are closer to
                > each other than to logical BAs.

                Initially I was sceptical that these constituted distinct notions---my
                reaction was, aren't algebraic, order, and logical BAs all the same?
                (Topological BAs are different of course, being those BAs with a
                topology and continuous homomorphisms.)

                However if one views an order BA as a poset with the requisite sups and
                infs satisfying the requisite laws, taking the signature to be that of a
                poset with no lattice operations, one could reasonably argue that the
                morphisms should be just all monotone functions. Is there any other
                notion of order BA that gives it a different extension from the
                algebraic or ordinary BAs?

                In the same vein, one could reasonably take the logical BAs to be either
                the initial BA (the prototypical BA defining the equational theory) or
                the free BAs (although one might question the value to logic of the
                uncountably generated free BAs).

                The method in the apparent madness of the different rules for
                topological and order BAs (morphisms respecting the meets and joins of
                the former but not the latter) can be illustrated with the example of
                CABAs, the complete atomic BAs. Since CABAs are automatically
                topological BAs, the above reasoning distinguishing order BAs from
                ordinary BAs when applied to CABAs should result in topological CABAs
                being distinguished from ordinary CABAs by taking their signature to
                exclude the sups and infs and to contain only the topology and the order
                (the latter being subsumable by the topology only if one gives up the
                convention that a CABA is a totally order disconnected space) and hence
                the morphisms to be the continuous monotone functions. Topological
                CABAs follow a different rule than topological BAs for the same reason
                order BAs follow a different rule than ordinary BAs: if they followed
                the same rule topological CABAs would merely be CABAs, just as order BAs
                would merely be BAs, and hence a waste of a potentially useful term.

                None of this however seems to support Jorge's sense that the algebraic,
                order, and topological BAs form a natural cluster separate from the
                logical BAs. On the other hand one can at least make the argument that,
                among all adjunctions F |- U for which UF is the monad (equational
                theory) of Boolean algebras, the logical and ordinary BAs are at the
                distal end from Set of the initial and final such adjunctions, being
                respectively the Kleisli and Eilenberg-Moore categories. In that sense
                they are as far apart as they can get. Absent an equally compelling
                force separating order and topological BAs from ordinary BAs, perhaps
                the natural gravity of mathematics pulls these three bodies into a tight
                but chaotic orbit around each other.

                We now know by hard experience to check the radars of journals. Lattice
                theory being completely off that of the Journal of Irreproducible
                Results, it would be irresponsible to waste its referees' time by
                submission of any substantive development of the above theme. As Arthur
                C. Clarke would have put it, any sufficiently advanced mathematics is
                indistinguishable from gibberish.

                Vaughan Pratt
              • Friedrich Wehrung
                Dear all, I was recently given the following link, which I read with much enjoyment: http://www.math.rutgers.edu/~zeilberg/Opinion81.html Many congratulations
                Message 7 of 7 , Aug 26, 2007
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                  Dear all,

                  I was recently given the following link, which I read with much enjoyment:


                  Many congratulations again to our 4-member Dilworth committee!

                  Cheers, Fred

                  --
                  Friedrich Wehrung
                  LMNO, CNRS UMR 6139
                  Universit\'e de Caen, Campus 2
                  D\'epartement de Math\'ematiques, BP 5186
                  14032 Caen cedex
                  FRANCE

                  e-mail: wehrung@...
                  alternate e-mail: fwehrung@...




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